Bernoulli Differential Equations AP Calculus BC

Bernoulli Differential Equations A Bernoulli differential equation can is of the form where P and Q are continuous functions on a given interval. Note that if n = 0 or 1, then we have a linear differential equation. But what if n > 1? Our goal is to take this non-linear differential equation and turn it into a linear differential equation, so we can solve it.

Reduction Divide by yn  We’re going to use u-substitution, so let u = y1–n  So Substitute back into original equation  Multiply by (1 – n)  The differential equation is now linear (ugly, but linear)!

Example 1 Find the general solution to the differential equation 1) n = 2, so u = y1–2, or u = y–1. 2) Divide equation by y2 to get RHS to equal x  3) Find du/dx  4) Solve for dy/dx 

Example 1 (cont.) 5) Substitute into equation in #2  6) Now this is a linear differential equation, so solve using integrating factor.

Example 1 (cont.) 7) Multiply by x–1  8) Product Rule in reverse  9) Integrate  10) Multiply by x  11) Substitute back for y (Recall u = y–1) y–1 = –x2 + Cx 

Example 2 Find the general solution to

Example 3 – Initial Value Problem Solve the differential equation with the initial condition y(1) = 0.

Still Confused? Watch this video: https://www.youtube.com/watch?v=7MmhoqvM9_Q It helped me!